What is the difference between an arithmetic sequence and an arithmetic series?

A sequence is the ordered list of numbers themselves (e.g., 3, 7, 11, 15, ...), while a series is their sum (e.g., 3 + 7 + 11 + 15 + \ldots). Use the arithmetic sequence formula to find individual terms; use the arithmetic series formula to find a total.

Can the common difference be a fraction or a decimal?

Yes, absolutely. The value of d can be any real number: 0.5, -1.3, \frac{1}{4}, etc. Substitute it into the formula exactly as you would an integer, and the formula works the same way.

How do I find which position n a certain term is in?

Rearrange the formula: from a_n = a_1 + (n-1)d, solve for n to get n = 1 + \frac{a_n - a_1}{d}. Plug in the term value you are looking for, the first term, and the common difference.

Does the formula work for sequences that are decreasing or going backwards?

Yes, the formula handles decreasing sequences perfectly because d is negative. For example, if the sequence is 20, 18, 16, 14, ..., then d = -2, and the formula applies as usual.

Arithmetic Sequence

Find the nth term in a sequence with constant differences between consecutive terms, or determine how many steps it takes to reach a specific value.

a_n = a_1 + (n - 1)d

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What each symbol means

What Arithmetic Sequence takes

Symbol	Meaning
$a_n$	The nth term of the sequence; the value at position $n$ that you are solving for. Confusing it with $n$ itself or with $a_1$ will give you the wrong answer.
$a_1$	The first term in the sequence; the value where the sequence starts. If you use a different starting term, your position counting is off by one or more places.
$n$	The position or index of the term you want, starting from 1 (so $n = 1$ means the first term). Using $n = 0$ or forgetting that you count from 1 will shift your final answer by $d$ .
$d$	The common difference; the constant amount that each term increases (if $d > 0$ ) or decreases (if $d < 0$ ) from the previous term. The sign of $d$ is critical; a negative $d$ flips the entire sequence's direction.

When to use it

When you have a sequence with a constant difference between terms and need to find a specific term without listing them all.

Level

Usually taught in: Algebra II · Appears on: SAT, ACT

Worked examples

1. Find the 7th term of a sequence with small integers

Problem

Find the 7th term of the sequence 3, 7, 11, 15, ...

$a_1 = 3, \quad d = 4$
The first term is 3, and the common difference is $7 - 3 = 4$ .
$a_7 = 3 + (7 - 1) \cdot 4$
Substitute into the formula $a_n = a_1 + (n-1)d$ with $a_1 = 3$ , $d = 4$ , and $n = 7$ .
$a_7 = 3 + 6 \cdot 4$
Simplify $7 - 1 = 6$ .
$a_7 = 3 + 24$
Multiply $6 \cdot 4 = 24$ .
$a_7 = 27$
Add $3 + 24 = 27$ .

Answer: $a_7 = 27$

We used the arithmetic sequence formula to jump directly to the 7th term instead of writing out all seven numbers. Since the first term and common difference are known, substitution and arithmetic give us the answer.

2. Find a term in a decreasing sequence with fractions

Problem

An arithmetic sequence starts at 10 and decreases by 2.5 each term. What is the 5th term?

$a_1 = 10, \quad d = -2.5$
The first term is 10, and the common difference is negative because the sequence decreases: $d = -2.5$ .
$a_5 = 10 + (5 - 1) \cdot (-2.5)$
Substitute into the formula with $a_1 = 10$ , $d = -2.5$ , and $n = 5$ .
$a_5 = 10 + 4 \cdot (-2.5)$
Simplify $5 - 1 = 4$ .
$a_5 = 10 + (-10)$
Multiply $4 \cdot (-2.5) = -10$ .
$a_5 = 0$
Add $10 + (-10) = 0$ .

Answer: $a_5 = 0$

This problem requires handling a negative common difference and a fractional decrease. Students often drop the negative sign or miscalculate when $d$ is negative; the formula handles both naturally if you keep the sign.

3. Bake sale scenario — find daily brownie sales on day 6

Problem

A school bake sale sells 4 brownies on day 1. Each day they sell 3 more brownies than the previous day. How many brownies do they sell on day 6?

$a_1 = 4, \quad d = 3, \quad n = 6$
The first day (day 1) they sell 4 brownies, the common difference is 3 brownies per day, and we want the 6th day.
$a_6 = 4 + (6 - 1) \cdot 3$
Use the formula $a_n = a_1 + (n-1)d$ with the values identified above.
$a_6 = 4 + 5 \cdot 3$
Simplify $6 - 1 = 5$ .
$a_6 = 4 + 15$
Multiply $5 \cdot 3 = 15$ .
$a_6 = 19$
Add $4 + 15 = 19$ .

Answer: $a_6 = 19 \text{ brownies}$

Word problems require translating English descriptions into the variables $a_1$ , $d$ , and $n$ . Once you identify these, the formula yields the answer without manually listing all six days.

Common mistakes

Where Arithmetic Sequence usually goes wrong

The mistake	Why it is wrong	The fix
Writing $a_n = a_1 + n \cdot d$ instead of $a_n = a_1 + (n-1)d$ , forgetting the $(n-1)$ .	This counts one extra multiple of $d$ , so every answer is off by exactly $d$ .	Remember that the first term (at $n = 1$ ) should equal $a_1$ plus zero multiples of $d$ . Since $n = 1$ gives $(1-1)d = 0$ , the formula $a_n = a_1 + (n-1)d$ is correct.
Confusing $n$ with the value of the term, writing $a_n = n$ or $a_{10} = 10$ without using the formula.	The index $n$ and the term value $a_n$ are different things; $n$ is just the position, not the value.	Always substitute $n$ into the formula and compute; the term value depends on $a_1$ and $d$ , not on $n$ alone.
Using only one difference to determine $d$ ; for example, computing $a_2 - a_1$ and assuming this is the common difference without checking $a_3 - a_2$ .	A sequence might have one pair with a certain difference but not be arithmetic; you must verify the pattern is consistent.	Always compute at least two successive differences— $a_2 - a_1$ and $a_3 - a_2$ —to confirm they are equal before trusting the arithmetic sequence formula.

Tips and when to use something else

Always compute at least two consecutive differences in a new sequence to verify it is arithmetic before applying this formula.
If you need the sum of the first $n$ terms (not just a single term), use the Arithmetic Series formula instead: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$ .
For sequences where each term is a constant multiple of the previous term (not a constant sum), switch to the Geometric Sequence formula instead.
To find the position $n$ when you know the term value $a_n$ , rearrange: $n = 1 + \frac{a_n - a_1}{d}$ .

Related formulas

Arithmetic SeriesFind the sum of an arithmetic sequence without adding every term—just use the first term, last term, and how many terms there are.Geometric SequenceThe formula a_n = a_1 r^{n-1} finds any term in a geometric sequence, where each term is a constant multiple of the previous one.Sum of First n IntegersAdd all integers from 1 to n using the formula \frac{n(n+1)}{2} instead of manually summing—much faster for large numbers.

Frequently asked questions

What is the difference between an arithmetic sequence and an arithmetic series?: A sequence is the ordered list of numbers themselves (e.g., 3, 7, 11, 15, ...), while a series is their sum (e.g., $3 + 7 + 11 + 15 + \ldots$ ). Use the arithmetic sequence formula to find individual terms; use the arithmetic series formula to find a total.
Can the common difference be a fraction or a decimal?: Yes, absolutely. The value of $d$ can be any real number: $0.5$ , $-1.3$ , $\frac{1}{4}$ , etc. Substitute it into the formula exactly as you would an integer, and the formula works the same way.
How do I find which position $n$ a certain term is in?: Rearrange the formula: from $a_n = a_1 + (n-1)d$ , solve for $n$ to get $n = 1 + \frac{a_n - a_1}{d}$ . Plug in the term value you are looking for, the first term, and the common difference.
Does the formula work for sequences that are decreasing or going backwards?: Yes, the formula handles decreasing sequences perfectly because $d$ is negative. For example, if the sequence is 20, 18, 16, 14, ..., then $d = -2$ , and the formula applies as usual.

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Reviewed 2026-09-18